PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 789 



Hence 



(ABF) = AB (44) 



since the distances from P to A and B are zero. A£ F is fixed through- 

 out the discussion we may consider (ABF) or AB as a product of A 

 and B. Since it is a number we call it the scalar product. If A and 

 B are not unit points we shall use the notation AB for the product 

 (ABF). It is then the distance from A to B multiplied by the product 

 of the magnitudes of A and B. 



From the definition, if 



B + C = D, 



we have 



AB + AC = (ABF) + (ACF) 



= (ADF) 

 = AD. 



The scalar product of two points has then the following properties : 



AB = -BA 



A (B + C) = AB + AC. (45) 



These laws hold for both AB and AB. The two differ in the fact 

 that AB is a multiple quantity, AB a number. The first forms an as- 

 sociative product 



AB C =z A BC, 



while the second does not. 



If a, b and f are unit lines not passing through F, 



(abf ) = ^ -f bf + E 



Since the angles bf and fa are zero 



(abf) = ^. (46) 



The angle between two lines may then be considered as a scalar product 

 of those lines. If a and b are not of unit magnitude, we define ab by 

 equation (45). This product dual to AB has the following properties: 



ab = — ba (47) 



a (b + c) = ab + ac. 

 Equations (45) together with the fact that AB is a number suggests 



